On cubic hypersurfaces with vanishing hessian

We prove that for $N\leq 6$ an irreducible cubic hypersurface with vanishing hessian in $\p^N$ is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of {\it Special Perazzo Cubic Hypersurfaces}, which, a posteriori, exhaust the class of cubic hypersurfaces with vanishing hessian, not cones, for $N\leq 6$. Finally we show by pertinent examples the technical difficulties arising for $N\geq 7$.